Ward's divisibility conjecture for Griesmer codes

Let $C$ be a $[g_q(k,d),k,d]_q$ Griesmer code, where $q=p^f$.

Ward's divisibility conjecture. Suppose that $p^e\mid d$ and $p^e\geq q$. Then $p^{e+1}/q$ is a divisor of $C$.

This conjecture seeks to extend Ward's divisibility results for Griesmer codes over nonprime fields. It is motivated by the fact that divisibility over prime fields is stronger, while examples such as the hexacode show that $q^e$-divisibility need not hold when $q=p^f$ with $f>1$; its resolution is not specified here.